Only a limited number of them, five to be exact if the pet is taken up to Mega.
The Pet Hatching System is designed in such a way where it is possible to predict the talent your pet should learn based on previous hatches and previously known talents.
However, there are times when pets can learn an unwanted talent. This happens quite often too.
You have 10 "possible" talents (and 10 "possible" Derby Abilities) but at this time the most you can have unlocked / manifested is 5 which is one each time your pet levels up - Teen, Adult, Ancient, Epic, and Mega. Time was pets maxed out at Epic, so you could only have 4 manifested, but then they introduced Mega as a level so all pets could get 5 possible.
For what it's worth there is another level higher than Mega - Ultra - but once you hit Ultra you only get a jewel to socket and no additional talent or ability.
Also, although you can only manifest 5 out of the possible 10, it's possible that your pet can get one more talent (or two!) by socketing a jewel. That talent, however, has nothing to do with 10 possible talents your pet has except to say that you cannot add a jewel if your pet already has that talent manifested (cannot add,for example, a Pain Giver jewel if your pet already has Pain Giver).
